The minimum sum of the distances of a real object and a real image from a concave mirror of radius of curvature 20cm is:

Options

A. 10 cm
B. 40 cm
C. 20 cm
D. none of these

Answer

To solve the problem of finding the minimum sum of the distances of a real object and a real image from a concave mirror with a radius of curvature of 20 cm, we can follow these steps:

### Step 1: Understand the relationship between the radius of curvature and the focal length of the mirror.
The focal length \( f \) of a concave mirror is given by the formula:
\[ f = -\frac{R}{2} \]
where \( R \) is the radius of curvature. Given \( R = 20 \) cm, we find:
\[ f = -\frac{20}{2} = -10 \text{ cm} \]

### Step 2: Define the distances of the object and the image.
Let:
- \( u \) be the distance of the object from the mirror (which is negative for real objects in the mirror convention).
- \( v \) be the distance of the image from the mirror (which is also negative for real images).

### Step 3: Use the mirror formula.
The mirror formula relates the object distance \( u \), image distance \( v \), and focal length \( f \):
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
Substituting \( f = -10 \) cm, we have:
\[ -\frac{1}{10} = \frac{1}{u} + \frac{1}{v} \]

### Step 4: Rearrange the equation.
Rearranging gives:
\[ \frac{1}{u} + \frac{1}{v} = -\frac{1}{10} \]
This can be rewritten as:
\[ \frac{u + v}{uv} = -\frac{1}{10} \]
Multiplying through by \( uv \) gives:
\[ u + v = -\frac{uv}{10} \]

### Step 5: Express the sum \( u + v \) in terms of \( u \) and \( v \).
We want to minimize \( |u| + |v| \). Since both \( u \) and \( v \) are negative, we can express this as:
\[ |u| + |v| = -u - v \]

### Step 6: Find the minimum value.
To find the minimum value of \( |u| + |v| \), we can use the fact that the minimum occurs when \( |u| = |v| \). Let \( |u| = |v| = x \). Then:
\[ u + v = -2x \]
Substituting into the mirror equation:
\[ \frac{1}{-x} + \frac{1}{-x} = -\frac{1}{10} \]
This simplifies to:
\[ -\frac{2}{x} = -\frac{1}{10} \]
Thus:
\[ \frac{2}{x} = \frac{1}{10} \implies x = 20 \text{ cm} \]

### Step 7: Calculate the minimum sum of distances.
Now substituting back:
\[ |u| + |v| = 2x = 2 \times 20 = 40 \text{ cm} \]

### Final Answer:
The minimum sum of the distances of a real object and a real image from the concave mirror is:
\[ \boxed{40 \text{ cm}} \]